How old is your puppy?
What are her symptoms?
At 12 weeks old, she should not be losing her teeth yet. The incisors, the tiny teeth in front of the mouth, start to fall out at approximately 14-16 weeks, then the other teeth follow.
However, a greenie may be just too hard for her fragile puppy teeth. She could have even broken a baby tooth on the greenie. You should check her mouth out if the symptoms of pain persist, or if you aren't sure, you can have your vet check the mouth. A broken puppy tooth can be painful, and sometimes needs to be pulled out so that it doesn't continue to bother the pup until it would fall out naturally. I have seen pups get an infection from a broken puppy tooth, and it was quite painful, they run fevers, and require antibiotics.
The frozen cloth is a good idea in the mean time, and baby aspirin can be used in pups, but I would need to know specifically how much she weighs in order to help with a dose. I would not recommend giving an aspirin unless you know for sure she is having a problem with her teeth. If she has an upset tummy, the aspirin will make her worse. I do not recommend anbesol, as some dogs have reactions to the medication in this pain reliever.